In a scattering problem, the main objective is solving the Schrödinger equation
where H is the total Hamiltonian of the scattering system in the center of momentum, K is the kinetic energy and V is the potential energy. We seek for a solution ,
The solution can be decomposed
The solution of can be solve by Runge-Kutta method on the pdf
where and .
For , the solution of is
where and is the Riccati-Bessel function. The free wave function is
where is the Legendre polynomial.
Note that, if we have Coulomb potential, we need to use the Coulomb wave instead of free wave, because the range of coulomb force is infinity.
For , the solution of can be found by Runge-Kutta method, where R is a sufficiency large that the potential is effectively equal to 0. The solution of is shifted
where is the scattering matrix element, it is obtained by solving the boundary condition at . The scattered wave function is
put the scattered wave function and the free wave function back to the seeking solution, we have the
and the differential cross section
In this very brief introduction, we can see
- How the scattering matrix is obtained
- How the scattering amplitude relates to the scattering matrix
But what is scattering matrix? Although the page did not explained very well, especially how to use it.